Lecture 1: Events and Probability

MA2224 Data Analysis January 27, 2020

Syllabus Highlights

Textbook: Probability and Statistics for Engineers and Scientists, 9th Edition, by Walpole and others. Pretty easy to find a pdf of, primarily for homework problems.
Grade: Best of two possible formulas found in syllabus. 90 or above is an A, F is less than 55.
Exam dates:
- Exam 1: Monday, March 2nd
- Exam 2: Wednesday, April 15th
- Final: Announced later
Dont cheat
Lecture schedule found in the syllabus. My numbers wont match the syllabus numbers.
Homework will be assigned roughly each week in the announcements section of the class page. HW will not be collected.
7 Quizzes will happen to you during the semester at the end of class. They will be announced in advance and your best 5 will count towards your quiz grade. They will be very similar to homework problems. No make-up quizzes will be given.
Note the calculator restriction in the sylabus.
If you have some situation that requires a missed exam or special considerations then you need to contact the math department. I have no administrative powers.

Constantine Specific Rules

Office hours: After class in this class room.
Email: ac5597@nyu.edu. Cell number: (347)-284-8367. Text problems or questions.
If you plan to use a laptop for non-class related stuff please sit in the back of the room.
Dont text or email me to let me know you will be late or absent, I wont know who you are and cant do anything about it.
Any food brought into class is subject to inspection and potential 15% confiscation.
I may be like 15 minutes late some days if the F train is being difficult.
Lecture slides will be emailed to you before class in a “class version”, then updated with corrections and solutions by the weekend. I still do a lot of board writing so you should still take notes.
Slides should open in a browser, use your browsers settings to adjust the zoom. F11 should full screen it.
They will probably show up in your spam folder, so add me as a safe sender.

Course Purpose

Despite being called Data Analysis, this class is essentially a classic probability and statistics course. What this means is that the methods taught in the class are primarily intended for controlled experiments that you might do in a lab.

Though a distinction wont often be made between experiments, where variables can be manipulated by the researcher, and observational studies, where researchers have little or no control over variables, it should be kept in mind that anything non-experimental usually needs added layers of thought in the real world.

A more modern class titled Data Analysis would probably include some data collection and cleaning, machine learning and statistical software.

Guiding Question

As a long term guiding question for the class, you can think of our ultimate goal as finding answers to the question:

“How does what you see, compare to what you expect to see?”

Concrete examples of this would be: “You toss a coin 100 times and get 40 heads, is the coin fair?” or “You scored a 980 on the SAT is this a good?” or “there are 2 bedbugs on your subway car, is that a lot?”. All three questions have the following in common:

A data element (40 heads, 980 grade, 2 bedbugs). “What you see”

A known or unknown expectation (50 heads, typical SAT score, usual bedbugs in a subway).

A question of “comparison”.

Typically, a researcher will be interested in determining the expected or usual value/behavior or they will be looking to see if something has deviates from this expected behavior We spend like 70% of the course dealing with the expectation and comparison elements of this question. Then a single class reminding you how to calculate mean, median etc. and then put it all together to answer the question. I’ll mention this idea throughout the semester and it may be useful to return to this core question if you ever feel lost and try to place the material in this framework. To start, we need to figure out what we should expect to happen when we do things, which means we need to know how likely things are to happen, so we gotta start with probability.

Sample Spaces

If probability tries to answer the question “What will happen?” then you have to start by thinking about all the things that could happen. This is called the sample space.

Sample Space

The Sample Space is the set of all possible outcomes from some action. It is the collection of things that you might possibly see.

Example:

What would be the sample space of flipping a coin once?

S S = {H e a d s, T a i l s}

$SS=\{Heads,Tails\}$

S S = {H e a d s, T a i l s, S i d e}

$SS=\{Heads,Tails, Side\}$

S S = {H e a d s, T a i l s, S i d e, L o s t}

$SS=\{Heads,Tails, Side, Lost\}$

S S = {H e a d s, T a i l s, S i d e, L o s t, S t o l e n, H i t s y o u o n h e a d k i l l s y o u, B l a c k h o l e t a k e s i t}

$SS=\{Heads,Tails, Side, Lost, Stolen, Hits\,\, you\,\, on\,\, head\,\, kills\,\, you\,\, , Black\,\, hole\,\, takes\,\, it\}$

In actuality, a sample space is somewhat subjective, but in this class we make the standard assumption in math classes which is to keep it boring.

Sample Space Examples

Give the sample space for rolling a die 2 times. How many outcomes are there?
When shooting free throws you can either hit your shot ( $H$ ) or miss ( $M$ ). Give the sample space for attempting 3 free throws. How many outcomes?
Give the sample space for the height of a Tandon student. How many outcomes?
When stealing your roommate’s food you can either be caught ( $C$ ) or escape ( $E$ ). Give the sample space for attempting to steal food until you get caught or you escape 5 times. How many outcomes?
When texting your ex, they can either respond ( $R$ ) or ignore you ( $I$ ). Give the sample space for texting your ex once every 5 minutes until they respond. How many outcomes?
Give the sample space for the length of time it will take a student to finish an exam in a 2 hour class. How many outcomes?
Give the sample space for the number of lectures a student will attend until they withdraw from the course from a total of 28 lectures but they have to withdraw by lecture 18. How many outcomes?
Give the sample space for the length of time it will take to pay back your student loans. How many outcomes?

Sample Space Examples

$S S = {(1, 1), (1, 2), (1, 3), . . ., (6, 5), (6, 6)} n = 36$ $SS=\{(1,1),(1,2),(1,3),...,(6,5),(6,6)\} \hspace{0.5 in} n=36$
$S S = {(H H H), (H H M), (H M H), (H M M), (M M M), (M M H), (M H M), (M H H)} n = 8$ $SS=\{(HHH),(HHM),(HMH),(HMM),(MMM),(MMH),(MHM),(MHH)\} \hspace{0.1 in} n=8$
$S S = {h : h \geq 0} n = \infty$ $SS=\{h : h \geq 0\} \hspace{0.5 in} n=\infty$
$S S = {(C), (E C), (E E C), (E E E C), (E E E E C), (E E E E E)} n = 6$ $SS=\{(C),(EC),(EEC),(EEEC),(EEEEC),(EEEEE)\} \hspace{0.5 in} n=6$
$S S = {(R), (I R), (I I R), (I I I R), (I I I I R), (I I I I I R) . . .} n = \infty$ $SS=\{(R),(IR),(IIR),(IIIR),(IIIIR),(IIIIIR)...\} \hspace{0.5 in} n=\infty$
$S S = {t : 0 < t \leq 2} n = \infty$ $SS=\{t : 0 < t \leq 2\} \hspace{0.5 in} n=\infty$
$S S = {0, 1, 2, 3, . . ., 17} n = 18$ $SS=\{0,1,2,3,...,17\} \hspace{0.5 in} n=18$
$S S = {t : t \geq 0} n = \infty$ $SS=\{t : t \geq 0\} \hspace{0.5 in} n=\infty$

Types of Sample Spaces

From the examples above we can identify two types of sample spaces:

Discrete Sample Spaces

A Discrete Sample Space is a sample space with either finitely many outcomes or an “infinite list” of outcomes.

Continuous Sample Spaces

A Continuous Sample Space is a sample space with an equivalent numbers of outcomes as points in an interval of real numbers.

In the examples above, the time to finish the exam, time to pay off studen loans and height of a Tandon student are continuous. The others are distcrete.

Since we just need basic probability ideas to get into statistics we only focus on discrete sample spaces until we move onto random variables.

Events

I’ll generally refer to the actual elements inside the sample space as outcomes to distinguish them from events (which technically they are also). When you start combining outcomes then we start talking more officially about set theory.

Events

An Event is any subset of a sample space.

E \subseteq S S

$E \subseteq SS$

Empty Set

The Empty Set is the event containing no sample space outcomes. Notation:

\emptyset

$\varnothing$ or

{}

$\{\}$

Complement

The Complement of an event

E

$E$ , is the set of all outcomes not in

E

$E$ denoted

E^{'}

$E^{\prime}$ .

Events

Union

The Union of two events

E

$E$ and

F

$F$ , is the set of all outcomes from

E

$E$ and

F

$F$ combined (no duplicates), denoted

E \cup F

$E \cup F$ .

Intersection

The Intersection of two events

E

$E$ and

F

$F$ , is the set of all outcomes that

E

$E$ and

F

$F$ have in common, denoted

E \cap F

$E \cap F$ .

Mutually Exclusive

Two events

E

$E$ and

F

$F$ are called Mutually Exclusive if their intersection is empty.

E \cap F = \emptyset

$E \cap F = \varnothing$ .

Events Example

Example:

Take the Sample Space of some experiment to be $SS=\{1, 2, 4, 10, 12, 15\}$ and consider events $A=\{4, 10\}$ , $B=\{1,2,4,15\}$ and $C= \{12\}$ . Determine the elements of the following events.

$A \cup B$
$A \cap B$
$A^{\prime} \cap B$
$A^{\prime} \cup B^{\prime}$
$(C \cup A) \cap B$
$(C \cup B)^\prime \cup A$
$A \cup B \cup C$
Which pairs of the three events are mutually exclusive?

Events Example

Take the Sample Space of some experiment to be $SS=\{1, 2, 4, 10, 12, 15\}$ and consider events $A=\{4, 10\}$ , $B=\{1,2,4,15\}$ and $C= \{12\}$ . Determine the elements of the following events.

A \cup B = {4, 10, 1, 2, 15}

$A \cup B = \{4, 10, 1, 2, 15\}$

A \cap B = {4}

$A \cap B = \{4\}$

A^{'} \cap B = {1, 2, 15}

$A^{\prime} \cap B = \{1, 2, 15\}$

A^{'} \cup B^{'} = {1, 2, 12, 15, 10}

$A^{\prime} \cup B^{\prime} = \{ 1,2, 12, 15, 10\}$

(C \cup A) \cap B = {4}

$(C \cup A) \cap B = \{4\}$

(C \cup B)^{'} \cup A = {4, 10}

$(C \cup B)^\prime \cup A = \{4,10\}$

A \cup B \cup C = S S

$A \cup B \cup C = SS$

Which pairs of the three events are mutually exclusive?

A \cap C = \emptyset

$A \cap C=\varnothing$ and

B \cap C = \emptyset

$B \cap C = \varnothing$

Events Example

Example:

Take the Sample Space of some experiment to be $SS=\{0, 1, 3, A, B, E, F, 20, 30\}$ . Determine the elements of the following events.

$(evens)\,\, \cap (over\,\,20)$
$(odds)^\prime\,\, \cap (letters)$
Are the events $odds^\prime$ and vowels mutually exlcusive?
Are the events $(odds \,\, \cup under\,\,10)^\prime$ and $(vowels)^\prime \cap (above\,\,15)$ mutually exclusive?

(e v e n s) \cap (o v e r 20) = {30}

$(evens)\,\, \cap (over\,\,20) = \{30\}$

(o d d s)^{'} \cap (l e t t e r s) = {A, B, E, F}

$(odds)^\prime\,\, \cap (letters) = \{A,B,E,F\}$

Are the events

o d d s^{'}

$odds^\prime$ and

v o w e l s

$vowels$ mutually exlcusive? No,

o d d s^{'} \cap v o w e l s = {A, E}

$odds^\prime \cap vowels=\{A,E\}$

Are the events

(o d d s \cup u n d e r 10)^{'}

$(odds \,\, \cup under\,\,10)^\prime$ and

(v o w e l s)^{'} \cup (a b o v e 15)

$(vowels)^\prime \cup (above\,\,15)$ mutually exclusive? No.

[(o d d s \cup u n d e r 10)^{'}] \cap [(v o w e l s)^{'} \cap (a b o v e 15)] = {A, B, E, F, 20, 30} \cap {20, 30} = {20, 30}

$\left[(odds \,\, \cup under\,\,10)^\prime\right] \cap \left[(vowels)^\prime \cap (above\,\,15)\right] = \{A,B,E,F,20,30\} \cap \{20,30\} = \{20,30\}$

Introduction to Probability: Axioms

Throwing probability into the story is pretty easy. All you do is take your sample space outcomes and assign each a number…with some rules called axioms meaning you just accept them because you are paying to trust what I say.

Axioms of Probability (Discrete Sample Spaces)

Given a sample space

S S = {E_{1}, E_{2}, E_{3}, . ., E_{n}}

$SS=\{ E_1, E_2, E_3,..,E_n\}$ where

n

$n$ may be infinite, the probability of the outcome

E_{i}

$E_i$ is a number

P (E_{i})

$P(E_i)$ that satisfies the following axioms:

$0 \leq P (E_{i}) \leq 1$ $0 \leq P(E_i) \leq 1$
$\sum_{i = 1}^{n} P (E_{i}) = 1$ $\sum_{i=1}^{n} P(E_i) = 1$
$P (E_{i} \cup E_{j}) = P (E_{i}) + P (E_{j}) i \neq j$ $P(E_i \cup E_j) = P(E_i)+P(E_j) \hspace{0.3 in} i \neq j$

Meaning?

The first axiom just says that you can make the probability of something anything you want so long as you stay within $0$ and $1$ . So you can tell me the probability of heads on a coin toss is $75\%$ but you cant tell me it’s $-37\%$ or $251\%$ .

The second axiom says that once you set the probabilities for all outcomes in a sample space, you need those outcome probabilities to add up to $1$ otherwise you are missing an outcome. So $P(heads) = 30\%$ and $P(tailss) = 70\%$ is fine but $P(heads) = 30\%$ and $P(tailss) = 60\%$ is not as there is something missing.

The final axiom says that if you want the probability that either of $2$ outcomes occur you just add their individual probabilities.

Probability Examples

Example:

Consider the sample space $SS=\{E_1,E_2,E_3,E_4,E_5\}$ and suppose that $P(E_1)= 0.1$ , $P(E_3)= 0.25$ , $P(E_2)=2P(E_5)= 0.4$ .

Determine $P(E_4)$ .
Determine $P(E_1 \cup E_4)$ .
Determine $P(\{E_1, E_5\}^{\prime})$ .

0.1 + 0.4 + 0.25 + P (E_{4}) + 0.2 = 1

$0.1 + 0.4 + 0.25 + P(E_4) + 0.2 = 1$

P (E_{4}) = 1 - 0.2 - 0.1 - 0.4 - 0.25 = 0.05

$P(E_4) = 1 - 0.2 - 0.1 - 0.4 -0.25 = 0.05$

P (E_{1} \cup E_{4}) = P (E_{1}) + P (E_{4}) = 0.1 + 0.05 = 0.15

$P(E_1 \cup E_4)= P(E_1) + P(E_4) = 0.1 + 0.05 = 0.15$

P ({E_{1}, E_{5}}^{'}) = P ({E_{2}, E_{3}, E_{4}}) = 0.4 + 0.25 + 0.05 = 0.7

$P(\{E_1, E_5\}^{\prime}) = P(\{E_2, E_3,E_4\}) = 0.4 + 0.25 + 0.05 =0.7$

Lecture 1: Events and Probability

Syllabus Highlights

Constantine Specific Rules

Past Semesters

Past Semesters

Course Purpose

Guiding Question

Sample Spaces

Sample Space

Example:

Sample Space Examples

Sample Space Examples

Types of Sample Spaces

Discrete Sample Spaces

Continuous Sample Spaces

Events

Events

Empty Set

Complement

Events

Union

Intersection

Mutually Exclusive

Events Example

Example:

Events Example

Events Example

Example:

Introduction to Probability: Axioms

Axioms of Probability (Discrete Sample Spaces)

Meaning?

Probability Examples

Example: